Question about changing variables in $\iiint_W x+3yz^2dV$

Question

I've had a multivariable calculus exam with this problem, I needed to calculate following integral:

$$\iiint_W x+3yz^2dV$$ $W = \{ (x,y,z) \in \Bbb R^3 : z^2 \geq 9x^2 + 9y^2, 0 \leq z \leq 3, x \geq 0\}$

My question is, would it make sense to make this variable change?:

$x = r \sin \theta$

$y = r \cos \theta$

$z = z$

Boundaries for integrating would be:

$0 \leq z \leq 3$

$0 \leq r \leq 1$

$0 \leq \theta \leq \pi$

I've tried it like this, but when I try substituting in the integral and multiplying it by the Jacobian's determinant it doesn't seem right.

What would a best change of variables be? Any suggestions or ideas?

Thanks!

Answer 1

Hint. Your approach is correct, but it should be $3r\leq z\leq 3$ and $-\pi/2\leq \theta\leq \pi/2$. Hence $$\int_{r=0}^1 \int_{z=3r}^3\int_{\theta=-\pi/2}^{\pi/2} \left(r\cos(\theta)+3r\sin(\theta)z^2\right) d\theta\,dz\, rdr.$$ Can you take it from here?